Seminar 4 point d.
Agent 2 knows his type (crazy/normal), while Agent 1 believes that with probability 0.1 agent 2 is crazy
while with probability 0.9 believes agent 2 is normal.
Therefore, agent 1 thinks that with probability 0.1 she is going to play the game
Crazy:
A
B
A
-2,0
2,-1
B
-1,2
0,0
while with probability 0.9 she is going to play the game:
Normal:
A
B
A
-2,-2
2,-1
B
-1,2
0,0
If agent 2 is crazy, F is strictly dominant. If agent 2 is normal, she mixes iff p*=2/3
For agent 1:
EU(A)=0.1[-2]+0.9[-2q+2(1-q)]=-0.2+0.9[2-4q]
EU(B)=0.1[-1]+0.9[-q]=-0.1-0.9q
Mixed strategy equilibrium
In a mixed strategy equilibrium, EU(A)=EU(B) which yields q*=17/27=0.63...
so in equilibrium:
agent 1 mixes with probability p*=2/3
agent 2 if crazy plays A, if normal mixes with probability q*=17/27
we can therefore describe the equilibrium as [2/3, (A,17/27)]
Pure strategy equilibria
Let's consider pure strategies and remember that the NE is the intersection of best replies.
IN pure strategies agent 2 will play (A,A) or (A,B).
First case (A,A)
If agent 2 (normal) plays A (or equivalently q=1), then the best reply from agent 1 is to play B because
EU(B)=-1>EU(A)=-2, that is BR(A,A)=B
But, is A the best response for agent 2 (normal) to 1 playing B, that is BR(B)=(A,A)? From the matrix is
straightforward to see that this is the case.
Hence, we have an equilibrium in pure strategies where
Agent 1 plays B
agent 2 if crazy plays A, if normal plays A
we can therefore describe the equilibrium as [B,(A,A)]
Second case (A,B)
If agent 2 (normal) plays B (or equivalently q=0), then the best reply from agent 1 is to play A because
EU(A)=1.6>EU(B)=-0.1, that is BR(A,B)=A But, is B the best response for agent 2 (normal) to 1 playing
A, that is BR(A)=(A,B)? From the matrix is straightforward to see that this is the case.
Hence, we have a second equilibrium in pure strategies where
Agent 1 plays A
agent 2 if crazy plays A, if normal plays B
we can therefore describe the equilibrium as [A,(A,B)]
Exercise
Two people are involved in a dispute. Person 1 does not know whether person 2 is strong or weak. He assigns
probability a to person 2 being strong. Person 2 is fully informed. Each person can fight or yield. Preferences are
represented by expected values of a payoff function that assigns payoff 0 if he yields and payoff of 1 if he fights
and opponent yields.
If both fight, payoffs are (-1,1) if strong and (1,-1) if weak.
a) Formulate the Baynesian game.
b) Find the BNE for a <1/2 and a>1/2.
ANSWER a)
Strong:
F
Y
F
-1,1
1,0
Y
0,1
0,0
F
Y
F
1,-1
1,0
Y
0,1
0,0
Weak:
Notice that if agent 2 is strong, then it is dominant to fight.
Consider now the expected utility for agent 1:
EU(F)= a(-1)+(1-a)(q+(1-q))= -a+(1-a)=1-2a
EU(Y)=a(0)+(1-a)(0)=0
If a<1/2 then agent 1 Fights. But the best response to fight for the weak agent 2 is Y, so the equilibrium
is (F;(F,Y))
if a>1/2 then agent 1 Yields. The best response for the weak agent 2 is F, so the equilibrium is (Y;(F,F))
If a=1/2 then for every q that belongs to (0,1) , {(1/2,1/2),(F,(q,1-q))} is an equilibrium